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A major drawback to the basal body temperature (BBT) method of birth control is Question 1 options: it can produce night sweats all of these choices are correct it does not predict ovulation it requires a good knowledge of anatomy and physiology

Lab Assignment-09 Note: Create and save m-files for each problem individually. Copy all the m-files into a ‘single’ folder and upload the folder to D2L. Read chapters 2 and chapter 3.1-3.3 of the textbook (Introduction to MATLAB 7 for Engineers), solve the following problems in MATLAB. Given A= [■(3&-2&1@6&8&-5@7&9&10)] ; B= [■(6&9&-4@7&5&3@-8&2&1)] ; C= [■(-7&-5&2@10&6&1@3&-9&8)] ; Find the following A+B+C Verify the associative law (A+B)+C=A+ (B+C) D=Transpose(AB) E=A4 + B2 – C3 Find F, given that F = E-1 * D-1 – (AT) -1 Use MATLAB to solve the following set of equations 5x+7y + 9z = 12 7x- 4y + 8z = 86 15x- 9y – 6z = -57 Write a function that accepts temperature in degrees F and computes the corresponding value in degree C. The relation between the two is Aluminum alloys are made by adding other elements to aluminum to improve its properties, such as hardness or tensile strength. The following table shows the composition of five commonly used alloys, which are known by their alloy numbers ( 2024, 6061, and so on) [Kutz, 1999]. Obtain a matrix algorithm to compute the amounts of raw materials needed to produce a given amount of each alloy. Use MATLAB to determine how much raw material each type is needed to produce 1000tons of each alloy. Composition of aluminum alloys Alloy % Cu % Mg % Mn % Si % Zn 2024 4.4 1.5 0.6 0 0 6061 0 1 0 0.6 0 7005 0 1.4 0 0 4.5 7075 1.6 2.5 0 0 5.6 356.0 0 0.3 0 7 0

Gene therapy may be used in the future to fight cancer by inserting genes that Select one: fight off mutations of the patient’s DNA. produce radioactive isotopes. cause cell death. produce anticancer drugs. all of the above.

Chapter 10 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A One-Dimensional Inelastic Collision Block 1, of mass = 3.70 , moves along a frictionless air track with speed = 15.0 . It collides with block 2, of mass = 19.0 , which was initially at rest. The blocks stick together after the collision. Part A Find the magnitude of the total initial momentum of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: m1 kg v1 m/s m2 kg pi Part B Find , the magnitude of the final velocity of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: Part C What is the change in the two-block system’s kinetic energy due to the collision? Express your answer numerically in joules. You did not open hints for this part. ANSWER: pi = kg m/s vf vf = m/s K = Kfinal − Kinitial K = J Conservation of Energy Ranking Task Six pendulums of various masses are released from various heights above a tabletop, as shown in the figures below. All the pendulums have the same length and are mounted such that at the vertical position their lowest points are the height of the tabletop and just do not strike the tabletop when released. Assume that the size of each bob is negligible. Part A Rank each pendulum on the basis of its initial gravitational potential energy (before being released) relative to the tabletop. Rank from largest to smallest To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: m h Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Momentum and Kinetic Energy Consider two objects (Object 1 and Object 2) moving in the same direction on a frictionless surface. Object 1 moves with speed and has mass . Object 2 moves with speed and has mass . Part A Which object has the larger magnitude of its momentum? You did not open hints for this part. ANSWER: Part B Which object has the larger kinetic energy? You did not open hints for this part. ANSWER: v1 = v m1 = 2m v2 = 2v m2 = m Object 1 has the greater magnitude of its momentum. Object 2 has the greater magnitude of its momentum. Both objects have the same magnitude of their momenta. Object 1 has the greater kinetic energy. Object 2 has the greater kinetic energy. The objects have the same kinetic energy. Projectile Motion and Conservation of Energy Ranking Task Part A Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: H PSS 10.1 Conservation of Mechanical Energy Learning Goal: To practice Problem-Solving Strategy 10.1 for conservation of mechanical energy problems. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 that makes an angle of 45 with the vertical, steps off his tree limb, and swings down and then up to Jane’s open arms. When he arrives, his vine makes an angle of 30 with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan’s speed just before he reaches Jane. You can ignore air resistance and the mass of the vine. PROBLEM-SOLVING STRATEGY 10.1 Conservation of mechanical energy MODEL: Choose a system without friction or other losses of mechanical energy. m VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you’re trying to find. SOLVE: The mathematical representation is based on the law of conservation of mechanical energy: . ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model The problem does not involve friction, nor are there losses of mechanical energy, so conservation of mechanical energy applies. Model Tarzan and the vine as a pendulum. Visualize Part A Which of the following sketches can be used in drawing a before-and-after pictorial representation? ANSWER: Kf + Uf = Ki + Ui Solve Part B What is Tarzan’s speed just before he reaches Jane? Express your answer in meters per second to two significant figures. You did not open hints for this part. ANSWER: Assess Part C This question will be shown after you complete previous question(s). Bungee Jumping Diagram A Diagram B Diagram C Diagram D vf vf = m/s Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass , and the surface of the bridge is a height above the water. The bungee cord, which has length when unstretched, will first straighten and then stretch as Kate falls. Assume the following: The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant . Kate doesn’t actually jump but simply steps off the edge of the bridge and falls straight downward. Kate’s height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle. Use for the magnitude of the acceleration due to gravity. Part A How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn’t touch the water. Express the distance in terms of quantities given in the problem introduction. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Spinning Mass on a Spring An object of mass is attached to a spring with spring constant whose unstretched length is , and whose far end is fixed to a shaft that is rotating with angular speed . Neglect gravity and assume that the mass rotates with angular speed as shown. When solving this problem use an inertial coordinate system, as drawn here. m h L k g d = M k L Part A Given the angular speed , find the radius at which the mass rotates without moving toward or away from the origin. Express the radius in terms of , , , and . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C R( ) k L M R( ) = This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). ± Baby Bounce with a Hooke One of the pioneers of modern science, Sir Robert Hooke (1635-1703), studied the elastic properties of springs and formulated the law that bears his name. Hooke found the relationship among the force a spring exerts, , the distance from equilibrium the end of the spring is displaced, , and a number called the spring constant (or, sometimes, the force constant of the spring). According to Hooke, the force of the spring is directly proportional to its displacement from equilibrium, or . In its scalar form, this equation is simply . The negative sign indicates that the force that the spring exerts and its displacement have opposite directions. The value of depends on the geometry and the material of the spring; it can be easily determined experimentally using this scalar equation. Toy makers have always been interested in springs for the entertainment value of the motion they produce. One well-known application is a baby bouncer,which consists of a harness seat for a toddler, attached to a spring. The entire contraption hooks onto the top of a doorway. The idea is for the baby to hang in the seat with his or her feet just touching the ground so that a good push up will get the baby bouncing, providing potentially hours of entertainment. F x k F = −kx F = −kx k Part A The following chart and accompanying graph depict an experiment to determine the spring constant for a baby bouncer. Displacement from equilibrium, ( ) Force exerted on the spring, ( ) 0 0 0.005 2.5 0.010 5.0 0.015 7.5 0.020 10 What is the spring constant of the spring being tested for the baby bouncer? Express your answer to two significant figures in newtons per meter. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Shooting a ball into a box Two children are trying to shoot a marble of mass into a small box using a spring-loaded gun that is fixed on a table and shoots horizontally from the edge of the table. The edge of the table is a height above the top of the box (the height of which is negligibly small), and the center of the box is a distance from the edge of the table. x m F N k k = N/m m H d The spring has a spring constant . The first child compresses the spring a distance and finds that the marble falls short of its target by a horizontal distance . Part A By what distance, , should the second child compress the spring so that the marble lands in the middle of the box? (Assume that height of the box is negligible, so that there is no chance that the marble will hit the side of the box before it lands in the bottom.) Express the distance in terms of , , , , and . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). k x1 d12 x2 m k g H d x2 = Elastic Collision in One Dimension Block 1, of mass , moves across a frictionless surface with speed . It collides elastically with block 2, of mass , which is at rest ( ). After the collision, block 1 moves with speed , while block 2 moves with speed . Assume that , so that after the collision, the two objects move off in the direction of the first object before the collision. Part A This collision is elastic. What quantities, if any, are conserved in this collision? You did not open hints for this part. ANSWER: Part B What is the final speed of block 1? m1 ui m2 vi = 0 uf vf m1 > m2 kinetic energy only momentum only kinetic energy and momentum uf Express in terms of , , and . You did not open hints for this part. ANSWER: Part C What is the final speed of block 2? Express in terms of , , and . You did not open hints for this part. ANSWER: Ballistic Pendulum In a ballistic pendulum an object of mass is fired with an initial speed at a pendulum bob. The bob has a mass , which is suspended by a rod of length and negligible mass. After the collision, the pendulum and object stick together and swing to a maximum angular displacement as shown . uf m1 m2 ui uf = vf vf m1 m2 ui vf = m v0 M L Part A Find an expression for , the initial speed of the fired object. Express your answer in terms of some or all of the variables , , , and and the acceleration due to gravity, . You did not open hints for this part. ANSWER: Part B An experiment is done to compare the initial speed of bullets fired from different handguns: a 9.0 and a .44 caliber. The guns are fired into a 10- pendulum bob of length . Assume that the 9.0- bullet has a mass of 6.0 and the .44-caliber bullet has a mass of 12 . If the 9.0- bullet causes the pendulum to swing to a maximum angular displacement of 4.3 and the .44-caliber bullet causes a displacement of 10.1 , find the ratio of the initial speed of the 9.0- bullet to the speed of the .44-caliber bullet, . Express your answer numerically. You did not open hints for this part. ANSWER: v0 m M L g v0 = mm kg L mm g g mm mm (v /( 0 )9.0 v0)44 Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. (v0 )9.0/(v0 )44 =

ENGR 3300: Fluid Mechanics, Fall 2015 Assignment 3 Due: Friday, Oct. 2, 2015 Topics: Chapter 3 & 4 Solutions must be neatly written and must include the following steps (if applicable) to receive full credit. 1. Given: List all known parameters in the problem. 2. Find: List what parameters the problem is asking you to find. 3. Solution: List all equations needed to solve the problem, and show all your work. Draw any necessary sketches or free body diagrams. Circle or box your final answer, and make sure to include appropriate units in your final answer. Grading: 15 total points (10 points for completeness + 5 points for one randomly chosen problem graded for correctness) 1. Water flows at a steady rate up a vertical pipe and out a nozzle into open air. The pipe diameter is 1 inch and the nozzle diameter is 0.5 inches. (a) Determine the minimum pressure that would be required at section 1 (shown in the figure below) to produce a fluid velocity of 30 ft/s at the nozzle (section 2). (b) If the pipe was inverted, determine the minimum pressure that would be required at section 1 to maintain the 30 ft/s velocity at the nozzle. 2. Water flows from a large tank through a small pipe with a diameter of 5 cm. A mercury manometer is placed along the pipe. Assuming the flow is frictionless, (a) estimate the velocity of the water in the pipe and (b) determine the rate of discharge (i.e. volumetric flow rate) from the tank. 3. An engineer is designing a suit for a race car driver and wants to supply cooling air to the suit from an air inlet on the body of the race car. The air speed at the inlet location must be 65 mph when the race car is traveling at 230 mph. Under these conditions, what would be the static pressure at the proposed inlet location? 4. Air flows downward toward a horizontal flat plate. The velocity field is given by ? = (??! − ??!)(2 + cos ??) where a = 5 s-1, ω = 2π s-1, and x and y (measured in meters) are horizontal and vertically upward, respectively, and t is in seconds. (a) Obtain an algebraic equation for a streamline at t = 0. (b) Plot the streamline that passes through point (x,y) = (3,3) at this instant.

Chapter 12 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Spinning Grinding Wheel At time a grinding wheel has an angular velocity of 26.0 . It has a constant angular acceleration of 33.0 until a circuit breaker trips at time = 1.80 . From then on, the wheel turns through an angle of 432 as it coasts to a stop at constant angular deceleration. Part A Through what total angle did the wheel turn between and the time it stopped? Express your answer in radians. You did not open hints for this part. ANSWER: Part B At what time does the wheel stop? Express your answer in seconds. You did not open hints for this part. ANSWER: t = 0 rad/s rad/s2 t s rad t = 0 rad Part C What was the wheel’s angular acceleration as it slowed down? Express your answer in radians per second per second. You did not open hints for this part. ANSWER: An Exhausted Bicyclist An exhausted bicyclist pedals somewhat erratically when exercising on a static bicycle. The angular velocity of the wheels follows the equation , where represents time (measured in seconds), = 0.500 , = 0.250 and = 2.00 . Part A There is a spot of paint on the front wheel of the bicycle. Take the position of the spot at time to be at angle radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the wheel’s rotation. What angular displacement has the spot of paint undergone between time 0 and 2 seconds? Express your answer in radians using three significant figures. s rad/s2 (t) = at − bsin(ct) for t 0 t a rad/s2 b rad/s c rad/s t = 0 = 0 Typesetting math: 29% You did not open hints for this part. ANSWER: Part B Express the angular displacement undergone by the spot of paint at seconds in degrees. Remember to use the unrounded value from Part A, should you need it. Express your answer in degrees using three significant figures. You did not open hints for this part. ANSWER: Part C What distance has the spot of paint moved in 2 seconds if the radius of the wheel is 50 centimeters? Express your answer in centimeters using three significant figures. You did not open hints for this part. ANSWER: = rad t = 2 = d Typesetting math: 29% Part D Which one of the following statements describes the motion of the spot of paint at seconds? You did not open hints for this part. ANSWER: Flywheel Kinematics A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration . The flywheel is assumed to be at rest at time in Parts A and B of this problem. Part A Find the time it takes to accelerate the flywheel to if the angular acceleration is . Express your answer in terms of and . d = cm t = 2.0 The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is increasing. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is increasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is increasing. t = 0 t1 1 1 Typesetting math: 29% You did not open hints for this part. ANSWER: Part B Find the angle through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocity . Express your answer in terms of some or all of the following: , , and . You did not open hints for this part. ANSWER: Part C Assume that the motor has accelerated the wheel up to an angular velocity with angular acceleration in time . At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of . Find , the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of some or all of the following: , \texttip{\alpha }{alpha}, and \texttip{t_{\rm 1}}{t_1}. You did not open hints for this part. t1 = 1 1 1 t1 1 = 1 t1 −5 t2 1 Typesetting math: 29% ANSWER: Surprising Exploding Firework A mortar fires a shell of mass \texttip{m}{m} at speed \texttip{v_{\rm 0}}{v_0}. The shell explodes at the top of its trajectory (shown by a star in the figure) as designed. However, rather than creating a shower of colored flares, it breaks into just two pieces, a smaller piece of mass \large{\frac15m} and a larger piece of mass \large{\frac45m}. Both pieces land at exactly the same time. The smaller piece lands perilously close to the mortar (at a distance of zero from the mortar). The larger piece lands a distance \texttip{d}{d} from the mortar. If there had been no explosion, the shell would have landed a distance \texttip{r}{r} from the mortar. Assume that air resistance and the mass of the shell’s explosive charge are negligible. Part A Find the distance \texttip{d}{d} from the mortar at which the larger piece of the shell lands. Express \texttip{d}{d} in terms of \texttip{r}{r}. You did not open hints for this part. \texttip{t_{\rm 2}}{t_2} = s Typesetting math: 29% ANSWER: Kinetic Energy of a Dumbbell This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational kinetic energy. You are to find the total kinetic energy \texttip{K_{\rm total}}{K_total} of a dumbbell of mass \texttip{m}{m} when it is rotating with angular speed \texttip{\omega }{omega} and its center of mass is moving translationally with speed \texttip{v}{v}. Denote the dumbbell’s moment of inertia about its center of mass by \texttip{I_{\rm cm}}{I_cm}. Note that if you approximate the spheres as point masses of mass m/2 each located a distance \texttip{r}{r} from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by I_{\rm cm} = mr^2, but this fact will not be necessary for this problem. Part A Find the total kinetic energy \texttip{K_{\rm tot}}{K_tot} of the dumbbell. Express your answer in terms of \texttip{m}{m}, \texttip{v}{v}, \texttip{I_{\rm cm}}{I_cm}, and \texttip{\omega }{omega}. You did not open hints for this part. \texttip{d}{d} = Typesetting math: 29% ANSWER: Part B This question will be shown after you complete previous question(s). Unwinding Cylinder A cylinder with moment of inertia \texttip{I}{I} about its center of mass, mass \texttip{m}{m}, and radius \texttip{r}{r} has a string wrapped around it which is tied to the ceiling . The cylinder’s vertical position as a function of time is y(t). At time t = 0 the cylinder is released from rest at a height \texttip{h}{h} above the ground. Part A The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate \texttip{\omega }{omega} and \texttip{v}{v}, the velocity of the center of mass of the cylinder? \texttip{K_{\rm tot}}{K_tot} = Typesetting math: 29% Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity. Express \texttip{\omega }{omega} in terms of \texttip{v}{v} and other given quantities. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C Suppose that at a certain instant the velocity of the cylinder is \texttip{v}{v}. What is its total kinetic energy, \texttip{K_{\rm total}}{K_total}, at that instant? Express \texttip{K_{\rm total}}{K_total} in terms of \texttip{m}{m}, \texttip{r}{r}, \texttip{I}{I}, and \texttip{v}{v}. You did not open hints for this part. ANSWER: Part D \texttip{\omega }{omega} = \texttip{K_{\rm total}}{K_total} = Typesetting math: 29% Find \texttip{v_{\rm f \hspace{1 pt}}}{v_f}, the cylinder’s vertical velocity when it hits the ground. Express \texttip{v_{\rm f \hspace{1 pt}}}{v_f}, in terms of \texttip{g}{g}, \texttip{h}{h}, \texttip{I}{I}, \texttip{m}{m}, and \texttip{r}{r}. You did not open hints for this part. ANSWER: Kinetic Energy of a Rotating Wheel A simple wheel has the form of a solid cylinder of radius \texttip{r}{r} with a mass \texttip{m}{m} uniformly distributed throughout its volume. The wheel is pivoted on a stationary axle through the axis of the cylinder and rotates about the axle at a constant angular speed. The wheel rotates \texttip{n}{n} full revolutions in a time interval \texttip{t}{t}. Part A What is the kinetic energy \texttip{K}{K} of the rotating wheel? Express your answer in terms of \texttip{m}{m}, \texttip{r}{r}, \texttip{n}{n}, \texttip{t}{t} and, \texttip{\pi }{pi}. You did not open hints for this part. ANSWER: Finding Torque \texttip{v_{\rm f \hspace{1 pt}}}{v_f} = \texttip{K}{K} = Typesetting math: 29% A force \texttip{\vec{F}}{F_vec} of magnitude \texttip{F}{F} making an angle \texttip{\theta }{theta} with the x axis is applied to a particle located along axis of rotation A, at Cartesian coordinates (0, 0) in the figure. The vector \texttip{\vec{F}}{F_vec} lies in the xy plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the xy plane. A particle is located at a vector position \texttip{\vec{r}}{r_vec} with respect to an axis of rotation (thus \texttip{\vec{r}}{r_vec} points from the axis to the point at which the particle is located). The magnitude of the torque \texttip{\tau }{tau} about this axis due to a force \texttip{\vec{F}}{F_vec} acting on the particle is given by \tau = r F \sin(\alpha) = ({\rm moment \; arm}) \cdot F = rF_{\perp}, where \texttip{\alpha }{alpha} is the angle between \texttip{\vec{r}}{r_vec} and \texttip{\vec{F}}{F_vec}, \texttip{r}{r} is the magnitude of \texttip{\vec{r}}{r_vec}, \texttip{F}{F} is the magnitude of \texttip{\vec{F}}{F_vec}, the component of \texttip{\vec{r}}{r_vec} that is perpendicualr to \texttip{\vec{F}}{F_vec} is the moment arm, and \texttip{F_{\rm \perp}}{F_\perp} is the component of the force that is perpendicular to \texttip{\vec{r}}{r_vec}. Sign convention: You will need to determine the sign by analyzing the direction of the rotation that the torque would tend to produce. Recall that negative torque about an axis corresponds to clockwise rotation. In this problem, you must express the angle \texttip{\alpha }{alpha} in the above equation in terms of \texttip{\theta }{theta}, \texttip{\phi }{phi}, and/or \texttip{\pi }{pi} when entering your answers. Keep in mind that \pi = 180\;\rm degrees and (\pi/2) = 90\;\rm degrees . Part A What is the torque \texttip{\tau_{\rm A}}{tau_A} about axis A due to the force \texttip{\vec{F}}{F_vec}? Express the torque about axis A at Cartesian coordinates (0, 0). You did not open hints for this part. Typesetting math: 29% ANSWER: Part B What is the torque \texttip{\tau_{\rm B}}{tau_B} about axis B due to the force \texttip{\vec{F}}{F_vec}? (B is the point at Cartesian coordinates (0, b), located a distance \texttip{b}{b} from the origin along the y axis.) Express the torque about axis B in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. You did not open hints for this part. ANSWER: Part C What is the torque \texttip{\tau_{\rm C}}{tau_C} about axis C due to \texttip{\vec{F}}{F_vec}? (C is the point at Cartesian coordinates (c, 0), a distance \texttip{c}{c} along the x axis.) Express the torque about axis C in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. You did not open hints for this part. ANSWER: \texttip{\tau_{\rm A}}{tau_A} = \texttip{\tau_{\rm B}}{tau_B} = Typesetting math: 29% Part D What is the torque \texttip{\tau_{\rm D}}{tau_D} about axis D due to \texttip{\vec{F}}{F_vec}? (D is the point located at a distance \texttip{d}{d} from the origin and making an angle \texttip{\phi }{phi} with the x axis.) Express the torque about axis D in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. ANSWER: Torque Magnitude Ranking Task The wrench in the figure has six forces of equal magnitude acting on it. \texttip{\tau_{\rm C}}{tau_C} = \texttip{\tau_{\rm D}}{tau_D} = Typesetting math: 29% Part A Rank these forces (A through F) on the basis of the magnitude of the torque they apply to the wrench, measured about an axis centered on the bolt. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: The Parallel-Axis Theorem Typesetting math: 29% Learning Goal: To understand the parallel-axis theorem and its applications To solve many problems about rotational motion, it is important to know the moment of inertia of each object involved. Calculating the moments of inertia of various objects, even highly symmetrical ones, may be a lengthy and tedious process. While it is important to be able to calculate moments of inertia from the definition (I=\sum m_ir_i^2), in most cases it is useful simply to recall the moment of inertia of a particular type of object. The moments of inertia of frequently occurring shapes (such as a uniform rod, a uniform or a hollow cylinder, a uniform or a hollow sphere) are well known and readily available from any mechanics text, including your textbook. However, one must take into account that an object has not one but an infinite number of moments of inertia. One of the distinctions between the moment of inertia and mass (the latter being the measure of tranlsational inertia) is that the moment of inertia of a body depends on the axis of rotation. The moments of inertia that you can find in the textbooks are usually calculated with respect to an axis passing through the center of mass of the object. However, in many problems the axis of rotation does not pass through the center of mass. Does that mean that one has to go through the lengthy process of finding the moment of inertia from scratch? It turns out that in many cases, calculating the moment of inertia can be done rather easily if one uses the parallel-axis theorem. Mathematically, it can be expressed as I=I_{\rm cm}+md^2, where \texttip{I_{\rm cm}}{I_cm} is the moment of inertia about an axis passing through the center of mass, \texttip{m}{m} is the total mass of the object, and \texttip{I}{I} is the moment of inertia about another axis, parallel to the one for which \texttip{I_{\rm cm}}{I_cm} is calculated and located a distance \texttip{d}{d} from the center of mass. In this problem you will show that the theorem does indeed work for at least one object: a dumbbell of length \texttip{2r}{2r} made of two small spheres of mass \texttip{m}{m} each connected by a light rod (see the figure). NOTE: Unless otherwise noted, all axes considered are perpendicular to the plane of the page. Part A Using the definition of moment of inertia, calculate I_{\rm cm}, the moment of inertia about the center of mass, for this object. Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B Using the definition of moment of inertia, calculate I_{\rm B}, the moment of inertia about an axis through point B, for this object. Point B coincides with (the center of) one of the spheres (see the figure). Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. ANSWER: Part C Now calculate I_{\rm B} for this object using the parallel-axis theorem. Express your answer in terms of \texttip{I_{\rm cm}}{I_cm}, \texttip{m}{m}, and \texttip{r}{r}. ANSWER: I_{\rm cm} = I_{\rm B} = I_{\rm B} = Typesetting math: 29% Part D Using the definition of moment of inertia, calculate I_{\rm C}, the moment of inertia about an axis through point C, for this object. Point C is located a distance \texttip{r}{r} from the center of mass (see the figure). Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. ANSWER: Part E Now calculate I_{\rm C} for this object using the parallel-axis theorem. Express your answer in terms of \texttip{I_{\rm cm}}{I_cm}, \texttip{m}{m}, and \texttip{r}{r}. ANSWER: Consider an irregular object of mass \texttip{m}{m}. Its moment of inertia measured with respect to axis A (parallel to the plane of the page), which passes through the center of mass (see the second diagram), is given by I_{\rm A}=0.64mr^2. Axes B, C, D, and E are parallel to axis A; their separations from axis A are shown in the diagram. In the subsequent questions, the subscript indicates the axis with respect to which the moment of inertia is measured: for instance, I_{\rm C} is the moment of inertia about axis C. I_{\rm C} = I_{\rm C} = Typesetting math: 29% Part F Which moment of inertia is the smallest? ANSWER: Part G Which moment of inertia is the largest? ANSWER: I_{\rm A} I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} I_{\rm A} I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} Typesetting math: 29% Part H Which moments of inertia are equal? ANSWER: Part I Which moment of inertia equals 4.64mr^2? ANSWER: Part J Axis X, not shown in the diagram, is parallel to the axes shown. It is known that I_{\rm X}=6mr^2. Which of the following is a possible location for axis X? ANSWER: I_{\rm A} and I_{\rm D} I_{\rm B} and I_{\rm C} I_{\rm C} and I_{\rm E} No two moments of inertia are equal. I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} between axes A and C between axes C and D between axes D and E to the right of axis E Typesetting math: 29% Torque and Angular Acceleration Learning Goal: To understand and apply the formula \tau=I\alpha to rigid objects rotating about a fixed axis. To find the acceleration \texttip{a}{a} of a particle of mass \texttip{m}{m}, we use Newton’s second law: \vec {F}_{\rm net}=m\vec{a}, where \texttip{\vec{F}_{\rm net}}{F_vec_net} is the net force acting on the particle. To find the angular acceleration \texttip{\alpha }{alpha} of a rigid object rotating about a fixed axis, we can use a similar formula: \tau_{\rm net}=I\alpha, where \tau_{\rm net}=\sum \tau is the net torque acting on the object and \texttip{I}{I} is its moment of inertia. In this problem, you will practice applying this formula to several situations involving angular acceleration. In all of these situations, two objects of masses \texttip{m_{\rm 1}}{m_1} and \texttip{m_{\rm 2}}{m_2} are attached to a seesaw. The seesaw is made of a bar that has length \texttip{l}{l} and is pivoted so that it is free to rotate in the vertical plane without friction. You are to find the angular acceleration of the seesaw when it is set in motion from the horizontal position. In all cases, assume that m_1>m_2, and that counterclockwise is considered the positive rotational direction. Part A The seesaw is pivoted in the middle, and the mass of the swing bar is negligible. Find the angular acceleration \texttip{\alpha }{alpha} of the seesaw. Express your answer in terms of some or all of the quantities \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{l}{l}, as well as the acceleration due to gravity \texttip{g}{g}. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B In what direction will the seesaw rotate, and what will the sign of the angular acceleration be? ANSWER: Part C This question will be shown after you complete previous question(s). \texttip{\alpha }{alpha} = The rotation is in the clockwise direction and the angular acceleration is positive. The rotation is in the clockwise direction and the angular acceleration is negative. The rotation is in the counterclockwise direction and the angular acceleration is positive. The rotation is in the counterclockwise direction and the angular acceleration is negative. Typesetting math: 29% Part D In what direction will the seesaw rotate and what will the sign of the angular acceleration be? ANSWER: Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Pivoted Rod with Unequal Masses The figure shows a simple model of a seesaw. These consist of a plank/rod of mass \texttip{m_{\rm r}}{m_r} and length 2x allowed to pivot freely about its center (or central axis), as shown in the diagram. A small sphere of mass \texttip{m_{\rm 1}}{m_1} is attached to the left end of the rod, and a small sphere of mass \texttip{m_{\rm 2}}{m_2} is attached to the right end. The spheres are small enough that they can be considered point particles. The gravitational force acts downward. The magnitude of the acceleration due to gravity is equal to \texttip{g}{g}. The rotation is in the clockwise direction and the angular acceleration is positive. The rotation is in the clockwise direction and the angular acceleration is negative. The rotation is in the counterclockwise direction and the angular acceleration is positive. The rotation is in the counterclockwise direction and the angular acceleration is negative. Typesetting math: 29% Part A What is the moment of inertia \texttip{I}{I} of this assembly about the axis through which it is pivoted? Express the moment of inertia in terms of \texttip{m_{\rm r}}{m_r}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{x}{x}. Keep in mind that the length of the rod is 2x, not \texttip{x}{x}. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Weight and Wheel Consider a bicycle wheel that initially is not rotating. A block of mass \texttip{m}{m} is attached to the wheel and is allowed to fall a distance \texttip{h}{h}. Assume that the wheel has a moment of inertia \texttip{I}{I} about its rotation axis. Part A Consider the case that the string tied to the block is attached to the outside of the wheel, at a radius \texttip{r_{\mit A}}{r_A} \texttip{I}{I} = Typesetting math: 29% . Find \texttip{\omega _{\mit A}}{omega_A}, the angular speed of the wheel after the block has fallen a distance \texttip{h}{h}, for this case. Express \texttip{\omega _{\mit A}}{omega_A} in terms of \texttip{m}{m}, \texttip{g}{g}, \texttip{h}{h}, \texttip{r_{\mit A}}{r_A}, and \texttip{I}{I}. You did not open hints for this part. ANSWER: Part B Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius \texttip{r_{\mit B}}{r_B} . Find \texttip{\omega _{\mit B}}{omega_B}, the angular speed of the wheel after the block has fallen a distance \texttip{h}{h}, for this case. Express \texttip{\omega _{\mit B}}{omega_B} in terms of \texttip{m}{m}, \texttip{g}{g}, \texttip{h}{h}, \texttip{r_{\mit B}}{r_B}, and \texttip{I}{I}. \texttip{\omega _{\mit A}}{omega_A} = Typesetting math: 29% You did not open hints for this part. ANSWER: Part C Which of the following describes the relationship between \texttip{\omega _{\mit A}}{omega_A} and \texttip{\omega _{\mit B}}{omega_B}? You did not open hints for this part. ANSWER: A Bar Suspended by Two Vertical Strings A rigid, uniform, horizontal bar of mass \texttip{m_{\rm 1}}{m_1} and length \texttip{L}{L} is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance d < L/2 from the left end of the bar and is connected to the ceiling; string B is attached to \texttip{\omega _{\mit B}}{omega_B} = \omega_A > \omega_B \omega_B > \omega_A \omega_A = \omega_B Typesetting math: 29% the left end of the bar and is connected to the floor. A small block of mass \texttip{m_{\rm 2}}{m_2} is supported against gravity by the bar at a distance \texttip{x}{x} from the left end of the bar, as shown in the figure. Throughout this problem positive torque is that which spins an object counterclockwise. Use \texttip{g}{g} for the magnitude of the acceleration due to gravity. Part A Find \texttip{T_{\mit A}}{T_A}, the tension in string A. Express the tension in string A in terms of \texttip{g}{g}, \texttip{m_{\rm 1}}{m_1}, \texttip{L}{L}, \texttip{d}{d}, \texttip{m_{\rm 2}}{m_2}, and \texttip{x}{x}. You did not open hints for this part. ANSWER: Part B Find \texttip{T_{\mit B}}{T_B}, the magnitude of the tension in string B. Express the magnitude of the tension in string B in terms of \texttip{T_{\mit A}}{T_A}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{g}{g}. \texttip{T_{\mit A}}{T_A} = Typesetting math: 29% You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). Part D If the mass of the block is too large and the block is too close to the left end of the bar (near string B) then the horizontal bar may become unstable (i.e., the bar may no longer remain horizontal). What is the smallest possible value of \texttip{x}{x} such that the bar remains stable (call it \texttip{x_{\rm critical}}{x_critical})? Express your answer for \texttip{x_{\rm critical}}{x_critical} in terms of \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{d}{d}, and \texttip{L}{L}. You did not open hints for this part. ANSWER: Part E \texttip{T_{\mit B}}{T_B} = \texttip{x_{\rm critical}}{x_critical} = Typesetting math: 29% This question will be shown after you complete previous question(s). A Tale of Two Nutcrackers This problem explores the ways that torque can be used in everyday life. Case 1 To crack a nut a force of magnitude \texttip{F_{\rm n}}{F_n} (or greater) must be applied on both sides, as shown in the figure. One can see that a nutcracker only applies this force at the point in which it contacts the nut (at a distance \texttip{d}{d} from the nutcracker pivot). In this problem the nut is placed in a nutcracker and equal forces of magnitude \texttip{F}{F} are applied to each end, directed perpendicular to the handle, at a distance \texttip{D}{D} from the pivot. The frictional forces between the nut and the nutcracker are equal and large enough that the nut doesn’t shoot out of the nutcracker. Part A Find \texttip{F}{F}, the magnitude of the force applied to each side of the nutcracker required to crack the nut. Express the force in terms of \texttip{F_{\rm n}}{F_n}, \texttip{d}{d}, and \texttip{D}{D}. You did not open hints for this part. ANSWER: Typesetting math: 29% Case 2 The nut is now placed in a nutcracker with only one lever, as shown, and again friction keeps the nut from slipping. The top “jaw” (in black) is fixed to a stationary frame so that a person just has to apply a force to the bottom lever. Assume that \texttip{F_{\rm 2}}{F_2} is directed perpendicular to the handle. Part B Find the magnitude of the force \texttip{F_{\rm 2}}{F_2} required to crack the nut. Express your answer in terms of \texttip{F_{\rm n}}{F_n}, \texttip{d}{d}, and \texttip{D}{D}. You did not open hints for this part. ANSWER: \texttip{F}{F} = \texttip{F_{\rm 2}}{F_2} = Typesetting math: 29% Part C This question will be shown after you complete previous question(s). Precarious Lunch A uniform steel beam of length \texttip{L}{L} and mass \texttip{m_{\rm 1}}{m_1} is attached via a hinge to the side of a building. The beam is supported by a steel cable attached to the end of the beam at an angle \texttip{\theta }{theta}, as shown. Through the hinge, the wall exerts an unknown force, \texttip{F}{F}, on the beam. A workman of mass \texttip{m_{\rm 2}}{m_2} sits eating lunch a distance \texttip{d}{d} from the building. Part A Find \texttip{T}{T}, the tension in the cable. Remember to account for all the forces in the problem. Express your answer in terms of \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{L}{L}, \texttip{d}{d}, \texttip{\theta }{theta}, and \texttip{g}{g}, the magnitude of the acceleration due to gravity. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B Find \texttip{F_{\mit x}}{F_x}, the \texttip{x}{x}-component of the force exerted by the wall on the beam ( \texttip{F}{F}), using the axis shown. Remember to pay attention to the direction that the wall exerts the force. Express your answer in terms of \texttip{T}{T} and other given quantities. You did not open hints for this part. ANSWER: Part C Find \texttip{F_{\mit y}}{F_y}, the y-component of force that the wall exerts on the beam ( \texttip{F}{F}), using the axis shown. Remember to pay attention to the direction that the wall exerts the force. Express your answer in terms of \texttip{T}{T}, \texttip{\theta }{theta}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{g}{g}. ANSWER: \texttip{T}{T} = \texttip{F_{\mit x}}{F_x} = \texttip{F_{\mit y}}{F_y} = Typesetting math: 29% Pulling Out a Nail A nail is hammered into a board so that it would take a force \texttip{F_{\rm nail}}{F_nail}, applied straight upward on the head of the nail, to pull it out. (Take an upward force to be positive.) A carpenter uses a crowbar to try to pry it out. The length of the handle of the crowbar is \texttip{L_{\rm h}}{L_h}, and the length of the forked portion of the crowbar (which fits around the nail) is \texttip{L_{\rm n}}{L_n}. Assume that the forked portion of the crowbar is perfectly horizontal. The handle of the crowbar makes an angle \texttip{\theta }{theta} with the horizontal, and the carpenter pulls directly along the horizontal. Typesetting math: 29% Part A With what force \texttip{F_{\rm pull}}{F_pull} must the carpenter pull on the crowbar to remove the nail? Express the force in terms of \texttip{F_{\rm nail}}{F_nail}, \texttip{L_{\rm h}}{L_h}, \texttip{L_{\rm n}}{L_n}, and \texttip{\theta }{theta}. You did not open hints for this part. ANSWER: Now, imagine that \texttip{F_{\rm pull}}{F_pull} is not large enough to dislodge the nail. In other words, the nail stays in place, and, if the surface below the crowbar weren’t present, the crowbar would rotate around the point of contact with the nail. This makes it natural to take the pivot point to be the point where the crowbar is in contact with the nail. (But you are always free to choose the pivot point to be any fixed point, even one some distance from the object.) Part B What is the magnitude of the normal force that the surface exerts on the crowbar, \texttip{F_{\rm bar}}{F_bar}? Express your answer for the normal force in terms of \texttip{F_{\rm pull}}{F_pull}, \texttip{\theta }{theta}, \texttip{L_{\rm n}}{L_n}, and \texttip{L_{\rm h}}{L_h}. Take the upward direction to be positive. You did not open hints for this part. ANSWER: Three bars are shown in the figure. Both bars A and B have \texttip{F_{\rm pull}}{F_pull} acting on them in the horizontal direction. Bar C has \texttip{F_{\rm pull}}{F_pull} = \texttip{F_{\rm bar}}{F_bar} = Typesetting math: 29% \texttip{F_{\rm pull}}{F_pull} strictly perpendicular to the bar. \texttip{L_{\rm h}}{L_h}, \texttip{L_{\rm n}}{L_n}, and \texttip{\theta }{theta} are the same quantities in each case. Part C Let the magnitude of the torque about the bend in the crowbars be denoted \texttip{\tau _{\mit A}}{tau_A}, \texttip{\tau _{\mit B}}{tau_B} and \texttip{\tau _{\mit C}}{tau_C} for each of the three cases shown. Which of the following is the correct relationship between the magnitude of of the torques? You did not open hints for this part. ANSWER: Tipping Crane \tau_A > \tau_B > \tau_C \tau_A > \tau_B = \tau_C \tau_A = \tau_B = \tau_C \tau_A < \tau_B = \tau_C \tau_A < \tau_B < \tau_C \tau_A = \tau_B > \tau_C \tau_A = \tau_B < \tau_C Typesetting math: 29% Learning Goal: To step through the application of \Sigma \vec{\tau} = 0 to prevent a crane from tipping over. A crane of weight \texttip{W}{W} has a length (wheelbase) \texttip{c}{c}, and its center of mass is midway between the wheels (i.e., the mass of the lifting arm is negligible). The arm extending from the front of the crane has a length \texttip{b}{b} and makes an angle \texttip{\theta }{theta} with the horizontal. The crane contacts the ground only at its front and rear tires. Part A While watching the crane in operation, an observer mentions to you that for a given load there is a maximum angle \texttip{\theta _{\rm max}}{theta_max} between 0 \degree and 90 \degree that the crane arm can make with the horizontal without tipping the crane over. Is this correct? ANSWER: Part B Later that week, while watching the same crane in operation, a different observer mentions to you that there is a maximum load the crane can lift without tipping, and you can find that maximum load by observing the minimum angle \texttip{\theta _{\rm min}}{theta_min} that the crane arm makes with the horizontal. Is this correct? ANSWER: yes no Typesetting math: 29% Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G This question will be shown after you complete previous question(s). Part H yes no Typesetting math: 29% Notice that we have the weight of the crane exerting a torque about the front wheels of the same crane. To create a torque, a force must be present, so it would seem that somehow the weight of the crane is exerting a force upon its front wheels. However, the crane is one object, and it follows from Newton's laws that an object cannot exert a net force upon itself. This crane seems to be defying Newton's laws. What's going on here? ANSWER: Part I Assume you get a summer job as a crane operator. On the first day you are lifting a heavy piece of machinery. Even though you have the arm at 70^\circ above the horizontal, the crane begins to tip slowly forward. Consider the following possible actions: Release the brake on the lifting cable so that the load accelerates 1. downward. 2. Release the lifting arm so that \texttip{\theta }{theta} decreases rapidly and the load accelerates downward. 3. Increase \texttip{\theta }{theta} while simultaneously letting out the lifting cable so that the load accelerates downward. 4. Put the crane wheels in gear and accelerate the crane forward. None of these solutions is ideal, but which will have the short-term effect of restoring contact of the crane's rear wheels with the ground? ANSWER: Spinning Situations Suppose you are standing on the center of a merry-go-round that is at rest. You are holding a spinning bicycle wheel over your head so that its rotation axis is pointing upward. The wheel is rotating counterclockwise when observed from above. Newton's laws don't apply to torques. The rear wheels exert a downward force on the front wheels. The crane is not accelerating so forces don't matter. The earth exerts forces on the crane and the load. all but 1 all but 2 all but 3 all but 4 all of them Typesetting math: 29% For this problem, neglect any air resistance or friction between the merry-go-round and its foundation. Part A Suppose you now grab the edge of the wheel with your hand, stopping it from spinning. What happens to the merry-go-round? You did not open hints for this part. ANSWER: Twirling a Baton A majorette in a parade is performing some acrobatic twirlings of her baton. Assume that the baton is a uniform rod of mass 0.120{\rm kg} and length 80.0{\rm cm} . Part A Initially, the baton is spinning about a line through its center at angular velocity 3.00{\rm rad/s} . What is its angular momentum? Express your answer in kilogram meters squared per second. It remains at rest. It begins to rotate counterclockwise (as observed from above). It begins to rotate clockwise (as observed from above). Typesetting math: 29% You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. \rm kg \cdot m^2/s Typesetting math: 29%

Not too long ago, major manufacturer of smart phone sync cables wanted to produce one final production run of the old cable before starting production of a new cable. Because millions of old devices still require the old cables, the manufacturer was comfortable using previous demand as a starting point for determining how many cables to make, and what order for materials they should place. In the past, the company had produced135, 000 cables every 90 days, with a standard deviation of demand of 15,000 cables every 90 days. Every cables old earned $10 profit. The cables cost $20 to make. How many cables should they produce in that last production run?